We consider the half-linear second order differential equation which is viewed as a perturbation of the so-called Riemann-Weber half-linear differential equation

133 (2008) MATHEMATICA BOHEMICA No. 2, 187–195

A REMARK ON POWER COMPARISON THEOREM FOR HALF-LINEAR DIFFERENTIAL EQUATIONS

Gabriella Bognár,Ondřej Došlý, Brno (Received November 30, 2006)

Abstract. We consider the half-linear second order differential equation which is viewed as a perturbation of the so-called Riemann-Weber half-linear differential equation. We present a comparison theorem with respect to the power of the half-linearity in the equation under consideration. Our research is motivated by the recent results published by J. Sugie, N. Yamaoka, Acta Math. Hungar. 111 (2006), 165–179.

Keywords: Riemann-Weber half-linear equation, Riccati technique, power comparison theorem, perturbation principle, principal solution

MSC 2000: 34C10

1. Introduction

The oscillation theory of half-linear second order differential equations (1) (r(t)Φα(x^′))^′+c(t)Φα(x) = 0, Φα(x) :=|x|^α−1x, α >0,

has attracted considerable attention in the recent years, let us mention at least the books [1], [4] and the references given therein. It was shown that the linear Sturmian theory extends directly to (1) and hence this equation can be classified as oscillatory or nonoscillatory similarly to the linear case. Elbert and Mirzov with their papers [5] and [9] are usually regarded as pioneers of the half-linear oscillation theory.

The first author has been supported by the Hungarian Scientific Grant K 61620 and the second author has been supported by the Research Project MSM0021622409 of the Ministry of Education of the Czech Republic and the Grant A1163401/04 of the Grant Agency of the Czech Academy of Sciences.

In the majority of oscillation criteria for (1), this equation is regarded as a per- turbation of the (nonoscillatory) one term equation

(r(t)Φα(x^′))^′ = 0

and (non)oscillation criteria for (1) are formulated in terms of the asymptotic be- havior of the function c with respect to the function r. Roughly speaking, if the function c is “sufficiently positive” (“not too positive”) with respect to the func- tion r, equation (1) becomes oscillatory (remains nonoscillatory). A more general approach consists in regarding (1) as a perturbation of the equation of the same form (2) (r(t)Φα(x^′))^′+ ˜c(t)Φα(x) = 0

and formulating conditions for (non)oscillation in terms of the asymptotic behavior of the function (c−˜c)h^α+1, where his a certain distinguished solution of (2); we refer to [4, Sec. 5.2] and [3] for more details. This approach was used for the first time in the paper of Elbert [6], where (1) withr(t)≡1is viewed as a perturbation of the half-linear Euler equation with the critical coefficient

(3) (Φα(x^′))^′+ ΓαΦα(x) = 0, Γα:= α α+ 1

α+1

.

The half-linear version of the classical Sturmian comparison theorem concerns a pair of equations (1) and

(4) (R(t)Φα(x^′))^′+C(t)Φα(x) = 0 and states that under the inequalities

0< R(t)6r(t), c(t)6C(t)

(4) oscillates faster than equation (1). More precisely, between any two consecutive zeros of a nontrivial solution of (1) there is at least one zero of any nontrivial solution of (4). In our paper we are concerned with another type of comparison theorems, namely, with respect to the power α in (1). The basic statement along this line is established in [11] (in a more general setting of dynamic equations on time scales).

Under some additional assumptions (which are trivially satisfied for r(t) ≡ 1 as treated later in this paper), it states that if β > αand (1) is oscillatory, then the equation

(r(t)Φβ(x^′))^′+c(t)Φβ(x) = 0 is also oscillatory.

In our paper we are motivated by the results presented in the paper [13], where the equation

(5) (Φα(x^′))^′+ 1

t^α+1[Γα+γαδ(t)] Φα(x) = 0, γα:= α α+ 1

α

,

is compared with the equation of the same form, but withαreplaced byβ. We will recall the results of [13] in more detail in the next section. We are also motivated by the results of [2] where the equation

(6) (Φα(x^′))^′+h Γα

t^α+1 + γα

2t^α+1lg²t+c(t)i

Φα(x) = 0

is viewed as a perturbation of the half-linear Riemann-Weber equation (sometimes called the Euler-Weber equation)

(7) (Φα(x^′))^′+ 1 t^α+1

hΓα+ γα

2t^α+1lg²t

iΦα(x) = 0.

As the main result of our paper, we present a comparison theorem with respect to αfor equation (6).

2. Auxiliary results

First we turn our attention to Euler and Riemann-Weber half-linear differential equations which are treated in detail in the paper of Elbert and Schneider [8]. The half-linear Euler equation

(8) (Φα(x^′))^′+ λ

t^α+1Φα(x) = 0

is oscillatory if and only ifλ >Γα and the half-linear Riemann-Weber equation (7) with a parameterλinstead ofγαis oscillatory if and only if λ > γα.

The transformation of the independent variablex(t) =y(s), s= lgt, transforms equation (6) to the equation

(9) (Φα( ˙y))˙−αΦα( ˙y) +h

Γα+ γα

2s² +c(e^s)i

Φα(y) = 0, ˙ = d ds

and the functionv= Φα( ˙y/y)is a solution of the Riccati type differential equation

(10) v˙+ γα

2s²c(e^s) +Hα(v) = 0, where

Hα(v) =αh

|v|^(α+1)/α−v+ 1 α+ 1

α α+ 1

αi .

The following lemma plays an important role in our investigation; its proof can be found e.g. in [4, Theorem 2.2.1].

Lemma 1. Equation(9)is nonoscillatory if and only if there exists a continuously differentiable functionv which satisfies the Riccati inequality

(11) v˙+ γα

2s²c(e^s) +Hα(v)60 for larges.

We will also need the following statement concerning asymptotics of solutions of Riccati inequalities. It is taken from [12].

Lemma 2. Suppose that a differentiable functionξ satisfies the inequality

(12) ξ(s) +˙ Hα(ξ(s))60 for larget.

Thenξis nonincreasing and tends to γα ass→ ∞.

Now we recall the concept of the principal solution of a nonoscillatory half-linear equation and the concept of the minimal solution of the associated Riccati equation (13) w^′+c(t) +α(r(t))^−1/α|w|^(α+1)/α= 0

as introduced in [10] and later independently in [7]. Nonoscillation of (1) implies that there exists a solution of (13) which is defined on some interval [T,∞)and among all such solutions there exists aminimal onew, minimal in the sense that any other˜ solution of (13) satisfiesw(t)>w(t)˜ for larget. The principal solution of (1) is then defined as the associated solution of (1), i.e., it is given by the formula

˜

x(t) = exp Z t

Φ⁻_α¹w(s)˜ r(s)

ds

, whereΦ⁻_α¹ is the inverse function ofΦα.

The next statement is the comparison theorem for minimal solutions of a pair of Riccati equations. Its proof can be found in [7], see also [4, Theorem 4.2.2].

Lemma 3. Consider the pair of half-linear equations(1),(2). Let˜c(t)>c(t)for larget, suppose that(2)is nonoscillatory and denote byw,˜ u˜ the minimal solutions of(13)and of

u^′+ ˜c(t) +α(r(t))^−1/α|u|^(α+1)/α= 0, respectively. Thenu(t)˜ >w(t)˜ for larget.

We conclude this section with the main results of [13] which served as motivation for our paper. In these statements,δ(t)is a continuous function which is positive for larget.

Proposition 1. Let0< α < β.

(i) If equation(5)is nonoscillatory, then the equation

(14) (Φβ(x^′))^′+ 1

t^β+1[Γβ+γβδ(t)] Φβ(x) = 0 is also nonoscillatory.

(ii) If the equation

(15) (Φβ(x^′))^′+ 1

t^β+1[Γβ+νδ(t)] Φβ(x) = 0

is nonoscillatory for someν > γβ, then (5)is also nonoscillatory.

3. A power comparison theorem

In this section we present the main result of our paper—a comparison theorem with respect to the powerαin the perturbed Riemann-Weber half-linear differential equation.

Theorem 1. Let0< α < β and suppose that the equation (16) (Φβ(x^′))^′+ 1

t^β+1

hΓβ+ γβ

2 lg²t +νδ(t)i

Φβ(x) = 0 is nonoscillatory for someν > γβ.Then the equation

(17) (Φα(x^′))^′+ 1 t^α+1

hΓα+ µ

lg²t+γαδ(t)i

Φα(x) = 0

is nonoscillatory for

µ < µ^∗:= γα(2γβν−γ_β²) 2ν² .

P r o o f. The transformation of the independent variable x(t) =y(s), s= lgt, transforms (17) and (16) to the equations

(18) (Φβ( ˙y))˙−βΦβ( ˙y) +h

Γβ+ γβ

2s² +νδ(e^s)i

Φβ(y) = 0

and

(19) (Φα( ˙y))˙−αΦα( ˙y) +h Γα+ µ

s² +γαδ(e^s)i

Φα(y) = 0.

The Riccati equation associated with (18) is (withξ= Φβ( ˙y/y))

(20) ξ˙+Hβ(ξ) + γβ

2s² +νδ(e^s) = 0, where

Hβ(ξ) =βh

|ξ|^(β+1)/β−ξ+ 1 β+ 1

β β+ 1

βi .

We will show that solvability of (20) for larges, which is equivalent to nonoscillation of (17), implies solvability of the Riccati inequality

(21) η˙+Hα(η) + µ

s² +γαδ(e^s)60 providedµ < µ^∗.Hence, suppose that µ < µ^∗and denote

ε= 2ν²(µ^∗−µ) γαγβ(ν−γβ).

There existε1>0, ε2>0 such that(1−ε1) (1−ε2)²= 1−ε.Denote further b

ε=ε1(ν−γβ) 2γαγβ

, εe=ε2γβ, c= ν γα

and consider the functionη(s) = (ξ(s) +ν−γβ)/c.We will show that this function satisfies inequality (21). We haveξ=cη−ν+γβ and hence

˙ η=1

cξ˙=−1 c

hγβ

2s²+Hβ(cη−ν+γβ) +νδ(e^s)i

=−γαγβ

2νs² −γαδ(e^s)−Hα(η)−1

cHβ(cη−ν+γβ) +Hα(η).

We will estimate the last two terms in the previous computation as follows. Denote F(η) :=1

cHβ(cη−ν+γβ)−Hα(η).

Then by a direct computation we have (see also [13])

F(γα) = 0 =F^′(γα) and F^′′(γα) =ν−γb

γαγβ

.

Hence,

1

cHβ(cη−ν+γβ)−Hα(η)>ν−γβ

2γαγβ

−bε

(η−γα)² (22)

=ν−γβ

2γαγβ (1−ε1) (η−γα)²

for largessinceη(s)→γαass→ ∞. Next we estimate the differenceη−γα.To this end, we estimate the difference ξ−γβ. By Lemma 2 we haveξ(s)ցγβ as s→ ∞ and the comparison theorem for minimal solutions of Riccati equations (Lemma 3) yieldsξ(s)> u(s)for larges, whereuis the minimal solution of the equation

˙

u+Hβ(u) + γβ

2s² = 0.

We haveu(s) = ( ˙z(s)/z(s))^β, wherez is the principal solution of the equation (Φβ( ˙z))˙−βΦβ( ˙z) +h

Γβ+ γβ

2s²

iΦβ(y) = 0

and it is known (see [8]) that

u(s) =γβ+γβ

s +o1 s

ass→ ∞.

Hence

ξ(s)−γβ> u(s)−γβ>γβ−eε s = γβ

s (1−ε2) for larges. Consequently,

η−γα=γα

ν (ξ−γα)> γαγβ

νs

1− eε γβ

= γαγβ

νs (1−ε2). Using this estimate and (22) we obtain

1

cHβ(cη−ν+γβ)−Hα(η)> ν−γβ

2γαγβ (1−ε1)γ_α²γ_β²

ν²s² (1−ε2)²

= (ν−γβ)γαγβ

2ν²s² (1−ε) for larges.

Therefore,

˙

η=−γαδ(e^s)−γαγβ

2νs² −Hα(η)−1

cHβ(cη−ν+γβ) +Hα(η)

<−γαδ(e^s)−Hα(η)− 1 s²

γαγβ

2ν

h1 + ν−γβ

ν (1−ε)i

=−γαδ(e^s)−Hα(η)− 1 s²

γαγβ

2ν

2ν−γβ−ε(ν−γβ) ν

=−γαδ(e^s)−Hα(η)−γαγβ

2ν

2ν−γβ

s² +ε(ν−γβ)γαγβ

2ν²s²

=−γαδ(e^s)−Hα(η)−µ^∗+ (µ−µ^∗) s²

=−γαδ(e^s)−Hα(η)− µ s²,

and henceη really satisfies (21) which means that (19) and so also (16) are nonoscil-

latory.

4. Remarks

(i) Part (i) of Proposition 1 can be directly extended to a more general situation than that treated in Theorem 1. Consider the half-linear differential equation

L^(α)_n (x) := (Φα(x^′))^′+ 1 t^α+1

h

Γα+γα

2 Xn

k=1

1 lg²₁tlg²₂t . . .lg²_kt

i

Φα(x) = 0,

wherelg₁t= lgt,lg_k+1t= lg(lg_kt). The Euler equation (3) and the Riemann-Weber equation (7) are special cases of (4) withn= 0andn= 1, respectively. It is konwn that the equationL^(α)n (x) = 0 is nonoscillatory and that the constants Γα and ¹₂γα

are optimal in the sense that if we repalce one of them by a greater constant, then the equation becomes oscillatory. Based on this observation, part (i) of Proposition 1 can be reformulated as follows.

Let0< α < β and consider the pair of equations L^(α)_n (x) +γαδ(t)Φα(x) = 0, (23)

L^(β)_n (x) +γβδ(t)Φβ(x) = 0, (24)

whereδis a nonnegative function. If equation(23)is nonoscillatory, then(24)is also nonoscillatory.

The proof of this statement easily follows from Proposition 1 (i) since (23) can be written in the form

(Φα(x^′))^′+ 1

t^α+1[Γα+γαδ(t)]Φ˜ α(x) = 0, where

˜δ(t) = 1 2

Xn

k=1

1

lg²₁lg²₂. . .lg²_kt +δ(t), and equation (24) can be rewritten in the same way.

(ii) An important part of the proof of Theorem 1 is the estimate ξ(s)−γβ = γβ/s+o(s⁻²)which we have used for the asymptotic formula

(25) 1

cHβ(cη(s)−ν+γβ)−Hα(η(s)) = ε(ν−γβ)γαγβ

2ν²s² +o(s⁻²)

as s → ∞. If we do apply this asymptotic formula and use the mere fact that the left-hand side of (25) is nonnegative (similarly as in [13]), we may reformulate Theorem 1 as follows:

Let 0< α < β and suppose that the equation L^(β)_n (x) +νδΦβ(x) = 0 is nonoscillatory for someν > γβ. Then the equation

L^(α)_n (x) +hγα(γβ−ν)

2νt^α+1lg²t +γαδ(t)i

Φα(x) = 0

is also nonoscillatory.

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Author’s address: Gabriella Bognár, Department of Mathematics, University of Miskolc, H-3514 Miskolc-Egytemváros, Hungary, e-mail: [email protected]; Ondřej Došlý, Department of Mathematics and Statistics, Masaryk University, Janáčkovo nám. 2a, CZ-602 00 Brno, Czech Republic, e-mail:[email protected].